Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#33428 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | #117 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C2×C8 | C24×C53 |
| socle | C2×C53 | C2×#432 |
| prime radical | C2×#41×C53 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C53×#405 ◁ G | |
528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,4,10,5,11,16,25,6,7,15,19,17,8,20,22)(3,9,13,24,14,23,28,32,12,18,26,29,27,21,30,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(2,7)(3,12)(4,15)(8,11)(14,27)(16,20)(24,29)(31,32);(2,8,7,11)(3,12)(4,15)(9,23,18,21)(10,25,19,22)(14,27)(16,20)(24,31,29,32);(1,3)(2,9)(4,13)(5,14)(6,12)(7,18)(8,21)(10,24)(11,23)(15,26)(16,28)(17,27)(19,29)(20,30)(22,31)(25,32)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 26416);